Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{z^2 - 9}{z + 3}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{9} = 3$ So we can rewrite the expression as: $t = \dfrac{({z} + {3})({z} {-3})} {z + 3} $ We can divide the numerator and denominator by $(z + 3)$ on condition that $z \neq -3$ Therefore $t = z - 3; z \neq -3$